Question

I’m still plugging away at the exercises in How to Design Programs on my own, but have managed to get stuck again. This time it’s question 11.4.7:

Develop the function
is-not-divisible-by<=i. It consumes a
natural number [>=1], i, and a natural
number m, with i < m. If m is not
divisible by any number between 1
(exclusive) and i (inclusive), the
function produces true; otherwise, its
output is false.

Use is-not-divisible-by<=i to define
prime?, which consumes a natural
number and determines whether or not
it is prime.

The first part I didn’t have too hard a time with:

;; A natural number [>=1] is either 1. 1 or 
;; 2. (add1 n) where n is a natural number [>=1].

;; is-not-divisible-by<=i : N[>=1] N -> boolean
(define (is-not-divisible-by<=i i m)
  (cond
    [(= i 1) true]
    [else (cond
            [(= (remainder m i) 0) false]
            [else (is-not-divisible-by<=i (sub1 i) m)])]))
(is-not-divisible-by<=i 3 6) ; expected: false.
(is-not-divisible-by<=i 6 7) ; expected: true.

But I just can’t see how I can use one variable to do this while using natural recursion. I thought about using a list, but that presents the same problems.

The only solution I can see is giving another variable–let’s say x– the same value as n, and then just doing it the way is-not-divisible-by<=i does it. But I think the authors intend for me to do it some other, simpler way. (And I don’t even know how to do what I described anyway, haha.)

This problem has really been busting my butt, so any help or hints, if you could, would be great!

Answer 1

A prime number is a number that cannot be divided by any number smaller than itself. You have already implemented the “cannot be divided by any number smaller than” part:

(define (prime? n)
  (is-not-divisible-by<=i (sub1 n) n))